The current governor of Texas is Rick Perry (Republican), in office since 2000. Governors are elected to terms of four years with no term limits. Until Perry, no governor had served more than seven-and-a-half consecutive years.
The Texas Legislature, like every state except Nebraska, has a bicameral legislature. This means there are two chambers: the House of Representatives and the Senate. The Texas House has 150 members–currently, 101 Republicans and 49 Democrats. Each is elected for a two-year term and represents roughly 113,000 people.I found some more information here. The Texas Senate has only 31 members–currently, 19 Republicans and 12 Democrats. Each senator is elected for a four-year term and represents close to 548,000 people. Neither chamber has a term limit.
The Legislature meets for 140 days every other year. When not in session, the governor can call the legislature back for a special session if necessary.
Texas’ “Supreme Court” is different from most states. The Texas Supreme Court itself only handles civil matters, including all juvenile cases. A separate court, the Texas Court of Criminal Appeals, handles all criminal matters of last resort. Both courts have nine justices each, elected to staggered six-year terms in partisan elections (all 18 at the moment are Republican).